Balwant Singh

Singh’s Basic Commutative Algebra is a very practical affair, inasmuch as it is explicitly geared toward applications to other parts of mathematics, eschews much of homological algebra (“… we do not define an additive or abelian category…”), and takes the reader/student from the basics of the theory of rings and ideals to some pretty advanced material.

The claim is that the book provides an introduction to commutative algebra at the postgraduate and research levels. The back-cover blurb concludes with “for the experienced researcher, the book may serve to present new perspectives on some well-known results, or as a reference.” This certainly does pigeonhole Basic Commutative Algebra as exactly what its title suggests, but with a definitive modern twist to it. In other words, Basic Commutative Algebra à la Singh is really rather different from, e.g., Atiyah-MacDonald’s Introduction to Commutative Algebra, which I used both in undergraduate and graduate school. I think it’s fair to say that the latter’s scope was narrower and emphatically oriented toward getting the fledgling graduate student ready for algebraic geometry. Singh, on the other hand, has a broader audience in mind and pushes (mildly) in the direction of research in commutative algebra itself. I don’t think either Atiyah or MacDonald had this in mind.

Another sine qua non among commutative algebra texts for people of my generation is, of course, the beefy contribution (in two volumes) Commutative Algebra by Oscar Zariski and Pierre Samuel. The authors’ names say it all, don’t they: the orientation is toward algebraic geometry, and the same distinction with Singh’s book can be drawn as was done vis à vis Atiyah-MacDonald (an exceptionally slim volume, by the way).

Thus, the book under review is indeed a departure from the aforementioned mainstays of the subject, even though it does quite properly lay out the foundations of the subject in the familiar manner: the theory of ideals is developed with the Zariski topology given its deserved airplay, followed by a treatment of modules and algebras including a discussion of exact sequences. It is this practical orientation to homological algebraic methods that is one of Basic Commutative Algebra’s primary virtues. Indeed the book’s Chapters 4, 5, 17, and 18 are expressly devoted to homological algebraic themes, and this is indeed very useful and meritorious in today’s mathematical climate.

Basic Commutative Algebra covers such standard important themes as Noetherian and Artinian conditions, primary decompositions, integrality, UFDs, DVRs, Dedekind domains, and the theory of valuations. The connections with algebraic number theory are given their proper due. Singh even goes beyond this in that Chapter 13 is devoted to transcendental extensions, culminating with Lüroth’s Theorem. In Chapter 14 we encounter both Noether normalization and Hilbert’s Nulstellensatz and in Chapter 15 we get to derivations and differentials. This is apt, given that two chapters later Singh does yeoman’s work on the topic of (co)homology, taking the reader from derived functors through resolutions to Ext and Tor and then to local cohomology and group (co)homology: that‘s a lot of serious stuff in a pretty compact orbit of about forty pages. I think this section is deserving of particular praise because it is clear that in this day and age every one interested even peripherally in algebraic x, where x = number theory, topology, geometry, for example, should be familiar with this material: Singh has performed quite a mitzvah in including it in his book in this way.

I guess the latter chapter of Basic Commutative Algebra disclose the author’s own algebraic proclivities (which are actually pretty orthodox): Cohen-Macualay theory, regular rings, and (well, well, well … ) stuff on the divisor class group. A lot of serious stuff in a pretty compact orbit, given that the book under review isn’t even 400 pages.

Pedagogically speaking, Basic Commutative Algebra also hits the mark: it’s easy to read, and contains good exercise sets. Caveat: Singh is anything but chatty, even though it’s not like you’re dealing with Landau’s notorious telegraph style of writing. Basic Commutative Algebra is clearly meant as a book to be worked through with commitment and effort. But doing so will yield valuable dividends.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.